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In Weil's short note entitled "On the Riemann hypothesis in function-fields" he mentions the notion of the complementary correspondence associated to a given correspondence $T:C\rightarrow C$ where $C$ is an algebraic curve. He gives a reference to one of Severi's papers for the definition, which I did not bother to look at since I probably won't be able to understand the definition.

Q1: So what is the definition of the complementary correspondence of $T$ in modern language?

Q2: Is there a good modern reference where the theory of correspondences is developed from scratch and is presented in a geometrical and intuitive way (so that it conveys the intuition of the Italian's school without distorting it too much) ?

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Given a correspondence $T \subset C \times C$ one defines an endomorphism of the Jacobian of $C$ by letting $T(P)$ be the divisor on $C$ cut out in $T$ by $\{P\}\times C$, extending by linearity and then taking linear equivalence classes. So the ring of endomorphisms is a quotient of the ring of correspondences. I think the complementary correspondence is a good representative of the equivalence class of the endomorphism $-T$ obtained by representing elements of the Jacobian as $\sum_{i=1}^g P_i - gP_0$, for a fixed $P_0$.

Modern books seem to focus on endomorphisms instead of correspondences (e.g. Mumford). There is a little bit of correspondences in Griffiths-Harris the way you want but it doesn't go very far.

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  • $\begingroup$ So if we think of $T:C\rightarrow C$ as a multivalued function and $T(P)=\sum_{i} [P_i]$ (formal sum) then are you saying that $(-T)(P)=\sum_{i} [-P_i]$? $\endgroup$
    – Hugo Chapdelaine
    Commented Aug 11, 2014 at 19:16
  • $\begingroup$ @HugoChapdelaine Yes. $\endgroup$
    – Felipe Voloch
    Commented Aug 11, 2014 at 19:19
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This is explained in no 17 of Igusa's paper "On the theory of algebraic correspondences...", J. Math. Soc. Japan 1 (1949), DOI: 10.2969/jmsj/00120147; it corresponds to Felipe Voloch's guess.

(There are many typos in Igusa's paper, and the Project Euclid scan does not improve them. Better read a hard copy if possible.)

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